Description: The definition of negation, in terms of -> and F. . (Contributed by Anthony Hart, 15-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | tbw-negdf | ⊢ ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.21 | ⊢ ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) | |
| 2 | ax-1 | ⊢ ( ¬ 𝜑 → ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) ) | |
| 3 | falim | ⊢ ( ⊥ → ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) ) | |
| 4 | 2 3 | ja | ⊢ ( ( 𝜑 → ⊥ ) → ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) ) |
| 5 | 4 | pm2.43i | ⊢ ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) |
| 6 | 1 5 | impbii | ⊢ ( ¬ 𝜑 ↔ ( 𝜑 → ⊥ ) ) |
| 7 | tbw-bijust | ⊢ ( ( ¬ 𝜑 ↔ ( 𝜑 → ⊥ ) ) ↔ ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ ) ) | |
| 8 | 6 7 | mpbi | ⊢ ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ ) |